The unimodular Spencer conjecture for complex vectors

Let $n\geq 1$ be an integer, and let $a_1,\ldots,a_n\in\mathbb{C}^n$ satisfy

$$\|a_i\|_{\infty}\leq 1\qquad (i=1,\ldots,n).$$

A complex number of modulus $1$ is unimodular, and a vector in $\mathbb{C}^n$ is a unimodular vector if all of its coordinates are unimodular. The unimodular Spencer conjecture. There exists a unimodular vector $x\in\mathbb{C}^n$ such that

$$|\langle x,a_i\rangle|\leq\sqrt{n}\qquad (i=1,\ldots,n).$$

This is the proposed complex analogue of Spencer's six standard deviations theorem. The source introduces it as the main conjecture, but the supplied text gives no resolution status.